Modify Equation

Algebra-1

1. Fundamental Concepts

Definition: An equation is a mathematical statement that asserts the equality of two expressions.
Variables: Symbols used to represent unknown quantities in equations.
Constants: Fixed numbers in an equation.
Equality: The symbol "=" indicating that the expressions on either side are equal.

2. Key Concepts

Basic Rule: $${\text{{Fe}}} \cdot x = y$$

Modification: To modify an equation, perform the same operation on both sides to maintain equality.

Application: Used to solve for unknown variables in various real-world problems.

3. Examples

Example 1 (Basic)

Problem: Solve the equation $$x + 5 = 12$$

Step-by-Step Solution:

Subtract 5 from both sides: $$x + 5 - 5 = 12 - 5$$
Simplify: $$x = 7$$

Validation: Substitute \(x = 7\) → Original: \(7 + 5 = 12\); Simplified: \(7 + 5 = 12\) ✓

Example 2 (Intermediate)

Problem: Solve the equation $$2y - 3 = 9$$

Step-by-Step Solution:

Add 3 to both sides: $$2y - 3 + 3 = 9 + 3$$
Simplify: $$2y = 12$$
Divide both sides by 2: $$\frac{2y}{2} = \frac{12}{2}$$
Simplify: $$y = 6$$

Validation: Substitute \(y = 6\) → Original: \(2(6) - 3 = 9\); Simplified: \(12 - 3 = 9\) ✓

4. Problem-Solving Techniques

Isolation Strategy: Isolate the variable on one side of the equation.
Balance Method: Perform the same operations on both sides to keep the equation balanced.
Check Solutions: Always substitute the solution back into the original equation to verify correctness.